If two equal chords of a circle intersect within the circle, prove that the segments of one chord are equal to corresponding segments of the other chord.

Let AB and CD be the two equal chords (AB = CD = a). Let the chords intersect at point P. Join OP. Draw OM and ON perpendicular to chords AB and CD respectively. We know that, Perpendicular from center bisects the chord. ∴ AM = MB = and CN = DN = Since, AB = CD. ∴ AM = MB = CN = DN = x(let) .....(1) In ∆ OMP and ∆ ONP, ⇒ ∠M = ∠N (Both equal to 90°) ⇒ OP = OP (Common side) ⇒ OM = ON (Equal chords are equidistant from the center.) ∴ ∆ OMP ≅ ∆ ONP (By R.H.S congruence rule) We know that, Corresponding parts of congruent triangles are equal. ∴ MP = NP = y(let) (By C.P.C.T.) .....(2) From figure, ⇒ CP = CN - NP = x - y and PB = MB - MP = x - y ∴ CP = PB. From figure, DP = CD - CP = a - (x - y) and AP = AB - BP = a - (x - y) ∴ AP = PD. Hence, proved that if two equal chords of a circle intersect within the circle, the segments of one chord are equal to corresponding segments of the other chord.

Prove that if chords of congruent circles subtend equal angles at their centres, then the chords are equal.

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Two circles of radii 5 cm and 3 cm intersect at two points and the distance between their centres is 4 cm. Find the length of the common chord.

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If two equal chords of a circle intersect within the circle, prove that the line joining the point of intersection to the centre makes equal angles with the chords.

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If a line intersects two concentric circles (circles with the same centre) with centre O at A, B, C and D, prove that AB = CD.

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Mathematics

If two equal chords of a circle intersect within the circle, prove that the segments of one chord are equal to corresponding segments of the other chord.

Circles

Related Questions

Prove that if chords of congruent circles subtend equal angles at their centres, then the chords are equal.

Two circles of radii 5 cm and 3 cm intersect at two points and the distance between their centres is 4 cm. Find the length of the common chord.

If two equal chords of a circle intersect within the circle, prove that the line joining the point of intersection to the centre makes equal angles with the chords.

If a line intersects two concentric circles (circles with the same centre) with centre O at A, B, C and D, prove that AB = CD.